Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3744 |
| automorphism | Aut(G) = AGL(1,ℤ/8)×AGL(2,𝔽3) (order 13824) · Inn(G) = C4⋊(C3⋊D3) · Out(G) = C22×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C32 ⋊ Q16 |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(20,21,22);(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(18,19)(21,22);(17,18,19)Polycyclic presentation:
2^4,3^2|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^239 conjugacy classes — view the full character table.
| maximal subgroup of | #63548, #63551, #63627, … |
| maximal normal subgroup of | #63548, #63551, #63627, … |
| maximal quotient of | #63643, #63645, C2×#3746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 89);